I'm definitely messing up somewhere :P(adsbygoogle = window.adsbygoogle || []).push({});

The Larmor Equation is

[tex]\Delta H = \frac{2\mu}{\hbar Mc^2 e} \frac{\partial V(r_{ij})}{\partial r_{ij}} ( L \cdot S)[/tex]

I derived an equation stated as

[tex]\nabla \times \vec{F}_{ij} \cdot \hat{n} = \nabla \times \frac{\partial V(r_{ij})}{\partial r_{ij}}[/tex]

How is put in a simpler way from the original derivation, is imagine we have what we began with

[tex]\vec{F}_{ij} = \frac{\partial V(r_{ij})}{\partial r_{ij}} \hat{n}[/tex]

Take the dot product of the unit vector on both sides gives

[tex]\vec{F}_{ij} \cdot \hat{n} = \frac{\partial V(r_{ij})}{\partial r_{ij}}[/tex]

Now taking the curl of F is

[tex]\nabla \times \vec{F}_{ij} \cdot \hat{n} = \nabla \times \frac{\partial V(r_{ij})}{\partial r_{ij}}[/tex]

In light of this equation however

[tex]\vec{F}_{ij} \cdot \hat{n} = \frac{\partial V(r_{ij})}{\partial r_{ij}}[/tex]

I could rewrite the Larmor energy as

[tex]\Delta H = \frac{2\mu}{\hbar Mc^2 e}(\vec{F}_{ij} \cdot \hat{n}) L \cdot S[/tex]

But I am sure many agree that is not very interesting.

(and this is really starting to make my brain cells burst), taking the curl of a force gives a F/length, however, the unit vector in this equation

[tex]\vec{F}_{ij} \cdot \hat{n} = \frac{\partial V(r_{ij})}{\partial r_{ij}}[/tex]

cancels these out and what I have again is the force again... yes? ... very circular... since this would be true, then we know what the force is anyway given earlier:

[tex]\vec{F}_{ij} = \frac{\partial V(r_{ij})}{\partial r_{ij}} \hat{n} [/tex]

Now, just a moment ago I found the Larmor energy written as

[tex]\Delta H = \frac{2\mu}{\hbar Mc^2 e} \frac{1}{r}\frac{\partial V(r_{ij})}{\partial r_{ij}} ( L \cdot S)[/tex]

Notice the 1/r term which is not in my original case. If that where true, and we plug in my force example again

[tex]\Delta H = \frac{2\mu}{\hbar Mc^2 e} \frac{1}{r}\frac{\partial V(r_{ij})}{\partial r_{ij}}\hat{n} ( L \cdot S)[/tex]

(If I am doing this right) the unit vector would cancel out with the radius term and what would be left with is

[tex]\Delta H = \frac{2\mu}{\hbar Mc^2 e} \frac{\partial V(r_{ij})}{\partial r_{ij}} ( L \cdot S)[/tex]

Now if that wasn't confusing I don't know what is.

I've either had it right from the beginning, or I have made a tiny mistake which is making a huge impact on my understanding of my own equation. Any help would be gladly appreciated.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Can you help me solve my mathematical problem?

**Physics Forums | Science Articles, Homework Help, Discussion**